Cao, J

Chang, WL

Guo, M

PolyU Institutional Repository

The Non-continuous Direction Vector I TestMinyi Guo1, Weng-Long Chang2 and Jiannong Cao31Department of Computer Software, The University of Aizu, Aizu-Wakamatsu City, Fukushima 965-8580, Japan 2Department of Management InformationSouthern Taiwan University of Technology, Tainan County, Taiwan 710, R.O.C. 3Department of ComputingHong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong E-mail:1minyi@u-aizu.ac.jp,2changwl@csie.ncku.edu.tw,3csjcao@comp.polyu.edu.hkAbstractIn this paper, we offer the non-continuous directionvector I test, an extension of the direction vector I test, to make sure whether there are integer-valued solutions forone-dimensional arrays with constant bounds andnon-one-increment.Index Terms ? Parallelizing Compilers, DataDependence Analysis.1. IntroductionThe data dependence problem in general case can bereduced to that of checking whether a system of one linear equation with m unknown variables has a simultaneousinteger solution, which satisfies the constraints for eachvariable in the system. Assume that a linear equations in a system is written as (1?1):whereeach is an integer for  and each  is ascalar integer variable for 1 Suppose that theconstraints to each variable in (1?1) are represented as (1?2):, X,0112211 aXaXaXaXa mmmm ????? ???ja mj ??0 X.mk ??kkk NXM ??kk = Mk + (m-1) * INCk and 1 ? m ? P,where  and  are integers for,kMmkN kINCk ??1  and  and  are, respectively,lower bound, upper bound and increment of a general loopand P is the number of loop iteration in the general loop andP = ,kM kN kINC.1)( ??ka ,1kkNINCM?Famous data dependencemethods include [1-9].ma1ma ,, 1?1 ?Xa??? ?mX 1?ma 1 mm Xa2. Background2.1. The Direction Vector I TestDefinitions 2?1, cited from [2, 9], defines an intervalequation.Definition 2?1: Let , L and U beintegers. A linear equation (2?1),22 ??Xa [L, U], is referred to as an interval equation.In light of [9], the direction vector I test considers a pairof same index variables to justify the movement of the twovariables to the right. A pair of same index variables in theequation (2?1) can be moved to the right if the coefficients ofthe two variables have small enough values to justify themovement of the two variables to the right.Proceedings of the 7th International Symposium on Parallel Architectures, Algorithms and Networks (ISPAN’04) 1087-4089/04 $ 20.00 © 2004 IEEE Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 4, 2009 at 03:21 from IEEE Xplore.  Restrictions apply.3. The Non-continuous Direction Vector I Test3.1.  Non-continuous Interval-EquationDefinition 3?0: Let [M, N, INC,INCMN ?+1] representthe non-continuous integer intervals from M to N, i.e., theset of all of the non-continuous integers,.11|)1(?????? ???????INCMNPINCPM ?Definition 3-1: Let be integers.For each k, 1? k ? m, let each of Mk and Nk be integer,where Mk ? Nk. If m > 0, then the equation,is said to be ([M1, N1, INC1,mm aaaa ,,,, 110 ??,0a?11 XaXa mm??111INCMN ?+1]; …; [Mm, Nm, INCm,mmMmINCN ?+1])-integer solvable if there exist integers j1, j2, …, jm, suchthat· .0112211 ajajajaja mmmm ????? ???· For each k, 1? k ? m: jk = Mk+(p-1)?INCk, where p is aninteger and 1 ? p ?kkkINCMN ?+1.?Definition 3-2: Let L and U beintegers. A non-continuous interval equation is an equationin the form of a [L, U, INC,mm aaaa ,,,, 110 ????? mm XaX 1 ?1INCLU ?11 ?Xa+1], which denotes the set of normal equationsconsisting of: L, …, L + (1 ?? ? mm Xa?1 XamX?? ? maINCLU ? ) ? INC = U.?Definition 3-3: Let L and U beintegers. For each k, 1? k ? m, let each of Mk and Nk be either an integer, where Mk ? Nk. If m > 0, then the non-continuous interval equation,,,,, 110 mm aaaa ????? mm XaXa 11 ?  [L, U, INC,INCLU ?+1] is said to be ([M1, N1, INC1,111INCMN ?+1];  …;[Mm, Nm,INCm,mmmINCMN ?+1])-integer solvable if one or more ofthe equations in the set which it denotes is ([M1, N1, INC1,111INCMN ?+1]; …; [Mm, Nm, INCm,mmmINCMN ?+1])-and|'{ sss'SS ???.|'{ sss'SS ???,{'21ssS ?,,,??[,,[1??LULni? sUsINCLUINCii,,,??[,,[1??LULni? sUsINCLUINCiiinteger solvable.?3.2. Mathematical PreliminariesDefinition 3-4: Let S and S  be sets of non-continuousintegers. We define an addition and a substitution operationon sets of non-continuous integer as follows:''s }'and SS ??}'and SS 's ?? Note that if S is thenon-continuous integer interval ]1, ?,,[ ?INCLUINCUL,,ns?and } , it follows that]1,]1??? '??INCLSUINC and ].1,]1??? '??INCLSUINC?Lemma 3-1: Let ]1,,, ??INCLUINCUL[  be aProceedings of the 7th International Symposium on Parallel Architectures, Algorithms and Networks (ISPAN’04) 1087-4089/04 $ 20.00 © 2004 IEEE Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 4, 2009 at 03:21 from IEEE Xplore.  Restrictions apply.non-continuous integer interval. Let]1,,,[ ??DIFMNDIFNMNDIFM ??be also a non-continuous integer interval, where . Let S = {b*y + c*z| y and z are,respectively, one element in ]1,,,[ ??DIFMNDIFNM????????|,*max(min(|*||,*max(|DIFbcDIFbt and y ? z}. Let (Part I): ???.0*if|)*)(||),*|0*if|)cbDIFcbDIFccbDIF]1)())()(),**))([]?????????????????????((*),,(*()1?????IN(**)((*,,,[????????CcbcLUcbDIFMDIFcMcbcbLSINCLUtINCbDIFINCNDIFINCLLUMNDIFcMcbUMNUINCULt ????0,INCINCLUt ??????INCofmultiple0,?tLUtaisand???]1()(*)((*)((*,,,[?????????????NLUMcbcbDIFMNUINCUL))()((*),),**))([?????????????INCcbcbINCDIFUDIFcMcbL,*()]1????????DIFMDIFcMNcbSINCLiff(Part II):iffINCtINCLUtINCLUtofmultipleaisand0, ???????.?)(11qqmnqqqqnqq ZcYbXa ?? ?????,,[ INCUL],1??INCLU,,,[ ???qqqqqqqINCMNINCNMXnq ??1for q ZY and]1,,,[ ??qqqqqqINCMNINCNM and qY ?.1for mqn ???qmnqqqqnqq cYbXa ?? ??????111()()([ mmmmm INCMNcbL ???????,**)( mmmmm INCcMcb ?????Proof: Omitted due to space limit.?3.3. Non-continuous Interval-Equation TransformationFirst, if two variables are related by a direction vectorconstraint of "=," they may be replaced by a single variable.Second, terms with zero coefficients may be omitted. Finally,a ">" constraint from one variable to another may be replacedby a constraint in the reverse direction. Taking all of thosepoints into account, we propose Lemma 3-2, which is extended from Theorem 3 in [9].Lemma 3-2: Let E=[(3-1), (3-2)], where (3-1) is equal to ,and (3-2) is equalto ]1 and ?q.qZLet E’=[(3-3), (3-4)], where (3-3) is equal to qZ)m()( mmmm INCMNcbU ????mmm Mcb *)( ??Proceedings of the 7th International Symposium on Parallel Architectures, Algorithms and Networks (ISPAN’04) 1087-4089/04 $ 20.00 © 2004 IEEE Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 4, 2009 at 03:21 from IEEE Xplore.  Restrictions apply.,,* INCINCc mm?],1))*)??INCm()((()(??????????INCcbcbMNLUmmmmmm,,[? qqq INCNMX,1for nq ? qY and,,,[ qqqNINCNMforand ? ZY qq????????????max(min(|*if*max(|*ifmmmmmm cbbcbtINCLUtm ???multipleaisand tmand (3-4) is equal to ]1, ??qqqqINCMN? ?qZ]1??qqqINCM1 ??? qn .1?m Let ? |)*)(||),*||,*.0|)*||,0mmmmmmmmmmINCcbINCcINCbINCcINCIf ,INCLUtm ????0,,of INC then E is integersolvable iff E’ is integer solvable.Proof: Omitted due to space limit.?We take an example to show the power of Lemmas3-1 and 3-2. Consider the normal linear equation (Ex1): X1- X2 = 0, subject to the constraints X1 and X2: [1, 9, 2, 5] andX1 < X2. First, the non-continuous direction vector I testtransforms the equation (Ex1) into the followingnon-continuous interval equation (Ex1-1): X1 - X2 = [0, 0, 2,1]. In light of Lemmas 3-1 and 3-2, because the coefficientsof X1 and X2 are, respectively, 1 and -1, t1 is equal to 2.Since t1 ? 2, 0 ? t1 ? 2 and t1 is a multiple of 2, thecondition of the movement for the pair of the same indexvariable, X1 and X2 is satisfied according to Lemma 3-2.Therefore, X1 and X2 are selected to move to the right-hand-side of (Ex1-1). Due to Lemma 3-2, a newnon-continuous interval equation is obtained (Ex1-2): 0 =[2, 8, 2, 4]. Because 2 ? 0 is false, 0 is not one element inthe non-continuous integer interval [2, 8, 2, 4]. Thus, thenon-continuous direction vector I test concludes that thereis no integer-valued solution.3.4. Interval-Equation Transformation Using the GCDTestIf all coefficients for variables in the non-continuousinterval equation have no sufficiently small values to justifythe movements of variables to the right, then Lemmas 3-1and 3?2 can not be applied to result in the immediatemovement. While every variable in a non-continuous intervalequation cannot be moved to the right, Theorem 3?1 andLemma 3-3 describe a transformation using the GCD test thatenables additional variables to be moved.Theorem 3-1: Let E=[(3-1), (3-2)], and let g = gcd(a1, ?,an, bn+1, ?, bm, cn+1, ?, cm). E is integer solvable iffg*?L/g? is one element of the integer set {L+(m-1)?INC| 1?m ?INCLU ?+1}.Proof: Omitted due to space limit.?Lemma 3-3: Let E=[(3-1), (3-2)], and let g=gcd . LetE’=[(3-5), (3-6)], where (3-5) is equal to),,,,,,,,(111 mnmnnccbbaa ??? ??)(1qqmnqqqq ZgcYgbX ?? ???1nqqga??,,[gUgL?Proceedings of the 7th International Symposium on Parallel Architectures, Algorithms and Networks (ISPAN’04) 1087-4089/04 $ 20.00 © 2004 IEEE Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 4, 2009 at 03:21 from IEEE Xplore.  Restrictions apply.],1, ??INCLUgINCand (3-6) is equal to ]1,,,[ ????qqqqqqqINCMNINCNMXnq ??1for  andand]1,,,[and ????qqqqqqqqINCMNINCNMZYforqq ZY ?  If L, U and INC are,respectively, a multiple of g then E is integer solvable iff E’is integer solvable..1 mqn ???Proof: Omitted due to space limit. ?3.5. Time ComplexityA pair of same index variables with small enoughcoefficients is easily found according to Lemmas 3?1 and3?2. In light of Lemmas 3?1 and 3?2, it is obvious that theworst-case time complexity to finding a pair of coefficientsenough is ?(m), where m is the number of variables in a non-continuous interval equation. The number of looking forall pairs of small enough coefficients in a non-continuousinterval equation is at most2m times because the numberof pairs moved in the non-continuous interval equation is at most2m pairs. Thus, the worst-case time complexity to move all pairs is ?(m2).To calculate the new non-continuous integer interval on the right-hand side of a non-continuous interval equation dueto the movement of the qualified pairs actually is equivalentto apply a single Banerjee-Wolfe inequality. Applying a single Banerjee-Wolfe inequality to calculate the lower bound and the upper bound of the new non-continuousinteger interval needs a constant time ?(y), where y is a constant. Thus, for calculating all new non-continuousinteger interval, the worst-case time complexity is ?(m)because there are at most2m  moves.If all coefficients in a non-continuous interval equationhave no absolute values of 1, then Lemma 3?3 employs theGCD test to reduce all coefficients to obtain small enoughcoefficients to justify the movement of a pair of same indexvariables to the right. In the worst cases, the non-continuousdirection vector I test contains m GCD tests. That study [2] shows that a large percentage of all coefficients haveabsolute values of 1 in one-dimensional array references withlinear subscripts in real programs. Therefore, the GCD test isnot always applied to reduce all coefficients in the equationsinferred from one-dimensional array references with linearsubscripts in real programs because all coefficients in theequations have at least an absolute value of 1. The worst-casetime complexity to the non-continuous direction vector I testto testing those one-dimensional array references with linear subscripts in real programs is immediately derived to be ?(m2). The worst-case time complexity of the directionvector I test is also ?(m2) [9]. Therefore, it is inferred that thenon-continuous direction vector I test still remains theefficiency of the direction vector I test.4. Experimental ResultsWe have tested our method and performed experimentson the codes abstracted from two numerical packages: Vector Loop and Livermore [10, 11]. 603 pairs of testedone-dimensional array references consisting of the same pairof array references with different direction vectors wereobserved under constant bounds and non-one-increment. TheProceedings of the 7th International Symposium on Parallel Architectures, Algorithms and Networks (ISPAN’04) 1087-4089/04 $ 20.00 © 2004 IEEE Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 4, 2009 at 03:21 from IEEE Xplore.  Restrictions apply.proposed method is only applied to test thoseone-dimensional arrays with subscripts under constantbounds and non-one-increment. It is very clear from Table 1that the proposed method could properly solve whether thereare definitive results for one-dimensional arrays withsubscripts under constant bounds and non-one-increment.BenchmarkThe number ofdefinitive resultsVector Loop 522Livermore 81Table 1. The result is to solve whether there areinteger-valued solutions for one-dimensional arrayswith subscripts under constant bounds andnon-one-increment.5. ConclusionsAccording to the time complexity analysis, theproposed method remains the efficiency of the directionvector I test. Therefore, assume that depending on theapplication domains and environments, the proposed methodcan be applied independently or together with other famousmethods to analyze data dependence for linear-subscriptarray references.References[1] Kleanthis Psarris, David Klappholz, and Xiangyun Kong. "On the Accuracy of the Banerjee Test," Journal of Paralleland Distributed Computing, 12(2), June 1991, pp. 152-158.[2] Xiangyun Kong, David Klappholz and Kleanthis Psarris."The I Test," IEEE Transaction on Parallel and DistributedSystems," Vol. 2, No. 3, July 1991, pp. 342?359.[3] Weng-Long Chang, Chih-Ping Chu and J. Wu, "AMulti-dimensional Version of the I Test," ParallelComputing, Vol. 27-13, Sept. 2001, pp. 1783-1799.[4] Weng-Long Chang, Chih-Ping Chu and J. Wu, "APrecise Dependence Analysis for Multi-dimensional ArraysUnder Specific Dependence Direction," Journal of Systemand Software. (Accepted, 2001).[5] Weng-Long Chang and Chih-Ping Chu. "TheGeneralized Direction Vector I Test," Parallel Computing,Vol. 27-8, July 2001, pp. 1117-1144.[6] Weng-Long Chang and Chih-Ping Chu. “The InfinityLambda Test: A Multi-dimensional Version of BanerjeeInfinity Test,” Parallel Computing, Vol. 26-10, Aug. 2000,pp. 1275-1295.[7] Weng-Long Chang, Chih-Ping Chu, and Jesse Wu. "TheGeneralized Lambda Test: A Multi-dimensional Version of Banerjee's Algorithm," International Journal of Parallel andDistributed Systems and Networks, Vol. 2, Issue 2, 1999,pp. 69-78.[8] Weng-Long Chang and Chih-Ping Chu. "The Extensionof the I Test," Parallel Computing, Vol. 24, Number 14,Nov. 1998, pp. 2101-2127.[9] Kleanthis Psarris, Xiangyun Kong, David Klappholz."The Direction Vector I test," IEEE Transaction on Paralleland Distributed Systems, Vol. 4, No. 11, 1993, pp.1280?1290.[10] David Levine, David Callahan and Jack Dongarra, "Acomparative study of automatic vectorizing compilers,"Parallel Computing 17(1991), pp.1223-1244.[11] W. Blume and R. Eigenmann. "Performance analysis ofparallelizing compilers on the perfect benchmarkprograms," IEEE Transaction on Parallel and DistributedSystems, Vol. 3, No. 6 (November 1992), pp. 643?656.s?Proceedings of the 7th International Symposium on Parallel Architectures, Algorithms and Networks (ISPAN’04) 1087-4089/04 $ 20.00 © 2004 IEEE Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on March 4, 2009 at 03:21 from IEEE Xplore.  Restrictions apply.

The non-continuous direction vector I test

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